AI 中文总结
本研究针对n维射影空间中连通射影曲线的邻域问题，构造了恰好带指定数目余维1全纯叶状结构的n维非紧复流形，确定其亚纯函数域的可能形式，推广并改进了曲面情形的相关结果。
AI 中文摘要
给定一条连通的射影曲线$C \subset \mathbb{P}^n$（其中$n \geq 2$）以及整数$0 \leq \ell \leq n$，我们构造了一个n维（非紧）复流形，它作为$C$的一个嵌入拷贝的邻域得到，其上恰好承载$\u2113$个余维为1的全纯叶状结构；此外，其上的每一个余维为1的分布都是这些叶状结构之一。我们还确定了这些流形的亚纯函数域：它可以被指定为$\mathbb{C}$或者超越次数为1的纯超越扩张，并且只要叶状结构的数量有限，就不可能有更大的域。这一结果将曲面上无叶状结构或无非常数亚纯函数的曲线邻域的先前构造推广到了任意维数，并对其进行了改进。
英文摘要
Given a connected projective curve $C \subset \mathbb{P}^n$, $n \geq 2$, and an integer $0 \leq \ell \leq n$, we construct an n-dimensional (non compact) complex manifold, obtained as a neighborhood of an embedded copy of $C$, which carries exactly $\ell$ codimension one holomorphic foliations; moreover, every codimension one distribution on it is one of these foliations. We also determine the field of meromorphic functions of these manifolds: it can be prescribed to be $\mathbb{C}$ or a purely transcendental extension of transcendence degree one, and no larger field is possible as soon as the number of foliations is finite. This extends to arbitrary dimension, and refines, previous constructions of neighborhoods of curves in surfaces without foliations or without non-constant meromorphic functions.
